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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 41405.f
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
41405.f1 | 41405g4 | \([1, -1, 1, -38673823, 92580237572]\) | \(11264882429818809/24990875\) | \(14191549494100125875\) | \([2]\) | \(1548288\) | \(2.9205\) | |
41405.f2 | 41405g2 | \([1, -1, 1, -2444448, 1412638322]\) | \(2844576388809/129390625\) | \(73476957439867515625\) | \([2, 2]\) | \(774144\) | \(2.5739\) | |
41405.f3 | 41405g1 | \([1, -1, 1, -415603, -74099294]\) | \(13980103929/3901625\) | \(2215612870494466625\) | \([2]\) | \(387072\) | \(2.2274\) | \(\Gamma_0(N)\)-optimal |
41405.f4 | 41405g3 | \([1, -1, 1, 1323407, 5371900356]\) | \(451394172711/22216796875\) | \(-12616235824153076171875\) | \([2]\) | \(1548288\) | \(2.9205\) |
Rank
sage: E.rank()
The elliptic curves in class 41405.f have rank \(0\).
Complex multiplication
The elliptic curves in class 41405.f do not have complex multiplication.Modular form 41405.2.a.f
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.